40th Canadian Mathematical Olympiad
Wednesday, March 26, 20081. ABCD is a convex quadrilateral for which AB is the longest side. Points M and N are located on sides AB and BC respectively, so that each of the segments AN and CM divides the quadrilateral into two parts of equal area. Prove that the segment M N bisects the diagonal BD.
2. Determine all functions f defined on the set of rational numbers that take rational values for which
f(2f(x) +f(y)) = 2x+y ,
for each x and y.
3. Let a,b,c be positive real numbers for which a+b+c= 1. Prove that
a−bc a+bc +
b−ca b+ca+
c−ab c+ab ≤
3 2 .
4. Determine all functionsfdefined on the natural numbers that take values among the natural numbers for which
(f(n))p ≡n mod f(p)
for all n ∈Nand all prime numbers p.
5. A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, i.e., the rook’s path is non-self-intersecting.
